= Sum of exponential variables with linearly increasing rates
{title2=$\sum_{i=1}^N T_i\overset d=\max_{i\leq N}E_i$}
If the <independent> <exponential random variables> $T_i$ have rates $i\lambda$, then
$$
\sum_{i=1}^N T_i\ \overset{d}{=}\ \max_{1\leq i\leq N}E_i,\qquad
\mathbb P\!\left(\sum_{i=1}^NT_i\leq t\right)=(1-e^{-\lambda t})^N\quad(t\geq0),
$$
where the $E_i$ are <independent> <exponential random variables> of rate $\lambda$. To see this, the first of $N$ independent exponential clocks rings at rate $N\lambda$; after it rings, the <memoryless property> leaves $N-1$ independent rate-$\lambda$ clocks. The successive <order statistic> gaps are therefore independent with rates $N\lambda,(N-1)\lambda,\ldots,\lambda$. Their sum has the same distribution as $\sum T_i$. In particular, for $x>0$, the tail at $Nx$ has logarithm $-\lambda Nx+o(N)$, since $1-(1-e^{-\lambda Nx})^N\sim N e^{-\lambda Nx}$. The sum divided by $N$ has a <large deviation principle> with speed $N$ and <rate function> $\lambda x$ on $x\geq0$, infinity otherwise.
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